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Probability is the measure of the likelihood that a particular event will occur. In the context of the emissions problem, we are interested in two probabilities:
- The probability that a single car emits more than 86 mg/mi
- The probability that the average emission level of 25 cars exceeds 86 mg/mi.
When we talk about probabilities in terms of a normal distribution, we refer to outcomes within the context of a bell curve, where certain values have specific probabilities of occurrence. This is crucial in determining how likely an emission level is to exceed specific limits. For our single car model, the computation involves locating where 86 mg/mi falls under this curve.
Similarly, when taking a fleet of 25 cars into account, the scenario shifts to evaluating how often their collective average would surpass the threshold, giving us insights into broader impacts.

A Z-score indicates how many standard deviations an element is from the mean of a distribution. It is calculated as follows:
\[ Z = \frac{X - \mu}{\sigma} \]
For individual (a) car emissions, X is 86 mg/mi, \(\mu\) is 80 mg/mi, and \( \sigma \) is 4 mg/mi. Calculating gives a Z-score of 1.5, which tells us that 86 mg/mi is 1.5 standard deviations above the average emission rate.
For (b) the fleet average emissions, we adjust the formula to use the standard error, if the sample size \(n\) is applied. In this case, a Z-score of 7.5 indicates that such a high average is extraordinarily unlikely under normal circumstances. All Z-scores provide a way to traverse probability tables and ascertain the likelihood of specific outcomes.

Standard deviation (SD) is a measure of variability or dispersion in a dataset. In simpler terms, it tells us how spread out numbers are around their average. A smaller standard deviation means data points are closer to the mean, while a larger one indicates they are spread out over a wider range.
In the exercise, the standard deviation of the emissions is given as 4 mg/mi. This helps us understand how consistent or variable the emission levels are from car to car. When calculating the Z-score, the standard deviation plays a crucial role in determining how unusual a data point (like an emission level) is in relation to the mean.
For groups, like the 25 cars, we use the standard deviation to compute the standard error, which gives us a precise understanding of the sample mean's variability, providing further insights into probability assessments.

Sample size is the number of observations or data points collected in a sample. The size of the sample significantly impacts the reliability of statistical measures, like the average, due to the law of large numbers, which suggests that larger samples produce more reliable estimates of the population mean.
In context (a) with just one car, we focus on individual emissions. But in context (b) when we consider 25 cars, we enter the realm of sampling distributions. Here, the sample size directly influences the calculation of the standard error, represented as:
\[ \sigma_{\bar{x}} = \frac{\sigma}{\sqrt{n}} \]
A larger sample size results in a smaller standard error, leading to more precise estimates of the population mean. Thus, the fleet's average emissions offer a different probability landscape due to the stabilizing effect of a larger (n) number.